This document provides an overview of geographic information systems (GIS) and topology. It discusses how GIS uses topology to represent spatial relationships between geometric objects and define properties like connectivity, area definition, and contiguity. The document outlines sources of errors in GIS data and describes techniques for building and validating topology to identify errors. It provides examples of topological rules and explains how Esri's ArcGIS software is used to create, validate, and fix topology errors in GIS vector data.

1. Lec #12
Fall 2013
LDA 150/ABT 150
4-Units
Introduction to
Geographic Information Systems (GIS)
Instructor:
Stavros G. Vougioukas, Ph.D.
Lecture: Tuesday and Thursday, 3 Kleiber, 3:10-4:30 PM
Lab Sections: 1137 PLESC Lab--Friday (01 & 04), Monday (02), Wednesday (03 & 06), Thursday (05)
11/6/2013 6:05 PM
1

2. Lecture contents
• Topology in GIS.
• Topology errors.
• Fixing topology.
11/6/2013 6:05 PM
2

3. Limitations of geometry
• Consider the geometric description of three
polylines.
– Both descriptions are geometrically equivalent.
(6.2,3.3)
(5.1,2.3)
1
Polyline
1
2
3
Coordinates
(6.6, 1.4), (6.2, 3.3), (5.1, 2.3), (4.8, 1.3)
(4.8, 1.3), (6.6, 1.4)
(4.8, 1.3), (5.7, 0.2), (6.6, 1.4)
(6.6,1.4)
2
(4.8,1.3)
3
(5.7,0.2)
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Polyline Coordinates
1
(4.8, 1.3), (5.1, 2.3), 6.2, 3.3), (6.6, 1.4)
2
(4.8, 1.3), (6.6, 1.4)
3
(4.8, 1.3), (5.7, 0.2), (6.6, 1.4)
3

4. Limitations of geometry
• If lines are a road network and polygons are
land parcels:
– What about one way streets?
– Which road is between parcels?
– Which parcel is on the ‘left’? (5.1,2.3)
(6.2,3.3)
1
• Something more than
geometry is needed…
(6.6,1.4)
b
a
2
(4.8,1.3)
11/6/2013 8:57 PM
3 (5.7,0.2)
4

5. Topology
• The representation of the arrangements of
geometric objects and the relationships among
them.
• Some fundamental relationships
– Connectivity: what is connected to what.
– Area definition: containment.
– Contiguity: borders and left/right adjacent areas.
11/6/2013 9:24 PM
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6. Topology
• Topological relationships are invariant under:
– Affine transformations
– Bending
– Stretching.
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http://www.indiana.edu/~gisci/courses/g338/lectures/introduction_vector.html

7. Directed graph (digraph)
• In GIS vector data topology is implemented
using graph theory.
• Graph: A set of
– nodes, and
– ordered pairs of nodes, called arcs
• Each arc connects two
nodes and has direction
from the 1st node to
the 2nd node.
11/7/2013 8:54 AM
7

8. Connectivity: Adjacency
• If an arc joins 2 nodes, the nodes are called
adjacent.
• Represented by the adjacency matrix.
– Encodes direction.
– No arc identity.
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9. Connectivity: Incidence
• If an arc joins 2 nodes, the nodes are called
incident to the arc.
• Incidence matrix
– Encodes direction, and
– Arc identity.
11/6/2013 10:35 PM
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10. ESRI Coverage: Topological properties
• Connectivity.
– Arcs connect to each other at nodes.
• Area definition.
– Connected arcs that surround an area define a
polygon.
• Contiguity.
– Arcs have direction and left and right sides).
11/7/2013 9:18 AM
Lo and Yeung p. 86
10

11. Coverage Connectivity: Arc-node topology
• Beginning and ending node of each arc.
• Remember: A node is either the end of a
line or the intersection of lines.
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12. Coverage Area definition:
Polygon-arc topology
• Area: series of connected
arcs; polygon-arc list.
• Containment:
– 0 precedes internal polygon.
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12

13. Coverage Contiguity: Left-right topology
• Relationships between arcs and their left and
right polygons.
• A background polygon (100) must be defined.
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14. Topology creation
• Can we build topology from geometry?
a b
1.
2.
Arc Coordinate List
Arc #
Coordinates
1 (4.8,1.3), (5.1,2.3), (6.2,3.3) (6.6, 1.4)
(6.6, 1.4)
2 (4.8,1.3), (6.2,3.3)
3 (6.2,3.3), (5.7,0.2), (4.8,1.3)
(6.6, 1.4)
(6.2,3.3)
1
(5.1,2.3)
(6.6,1.4)
b
a
2
(4.8,1.3)
3 (5.7,0.2)
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2. Arc-Node List
1.
Arc #
From-node To- node
1a
b
2a
b
3b
a
14

15. Topology creation
2. Arc-Node List
1.
Arc #
1
A
From-node To- node
1a
b
2a
b
3b
a
b
a
B
3
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2
3. Polygon-Arc List
Poly #
Arcs
A
1, 2
B
2, 3
15

16. Topology & errors
• Building topology from geometry involves
identifying all nodes, arcs, and polygons.
• What if there are errors in the geometry?
(6.19,3.3)
(6.2,3.3)
1
(5.1,2.3)
– No problem in topology…
–Only in geometry.
(6.6,1.4)
b
a
2
(4.8,1.3)
Polyline Coordinates
1
3 (5.7,0.2)
11/7/2013 10:41 AM
(4.8, 1.3), (5.1, 2.3), (6.19, 3.3), (6.6, 1.4)
2
3
(4.8, 1.3), (6.6, 1.4)
(4.8, 1.3), (5.7, 0.2), (6.6, 1.4)
16

17. Topology & errors
• What about this error?
Polyline Coordinates
(6.2,3.3)
(5.1,2.3)
1
1
2
3
(4.8, 1.3), (5.1, 2.3), 6.2, 3.3), (6.599, 1.4)
(4.8, 1.3), (6.6, 1.4)
(4.8, 1.3), (5.7, 0.2), (6.6, 1.4)
(6.599,1.4) – What happens to node b?
2
(6.6,1.4)
–What happens to polygons?
(4.8,1.3)
3
(5.7,0.2)
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17

18. Topology & errors
• Nonsensical vertex order (ambiguity).
Polyline Coordinates
(6.2,3.3)
(5.1,2.3)
1
1
2
3
(4.8, 1.3), (6.2, 3.3), (5.1, 2.3), (6.6, 1.4)
(4.8, 1.3), (6.6, 1.4)
(4.8, 1.3), (5.7, 0.2), (6.6, 1.4)
(6.6,1.4)
2
(4.8,1.3)
3
(5.7,0.2)
11/7/2013 11:46 AM
18

19. Geometry errors
• Geometrical errors
– May cause topological errors, i.e., violations of
topological relathionships
– May only affect spatial data accuracy.
• Accuracy standards govern ‘acceptable’
geometrical errors (see Book, section 7.2).
• Topological errors caused by very small
geometry errors can cause severe effects!
11/7/2013 10:31 AM
19

20. ‘Small’ error example
• An ambulance carrying a stroke victim needs
to take the fastest route from A to B.
• Two arcs in a road network do not ‘exactly
connect’.
• The car navigator takes a long route…
20 miles
B
A
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3 miles
14 miles
20

21. Line feature topological errors
• Undershoot
– Gap between lines.
• Overshoot
– Overextended lines.
• Both errors cause
dangling nodes.
– Such nodes could be OK
though (when?)
11/7/2013 11:16 AM
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22. Line feature topological errors
• Pseudonodes
– Nodes not located at end of arc or at line
intersection.
• Direction of line.
– Important in the case of
one way streets or rivers/streams.
11/7/2013 12:32 PM
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23. Polygon feature topological errors
• Unclosed polygons.
• Gap between two polygons.
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24. Polygon feature topological errors
http://scapetoad.choros.ch/help/v11/a-cartogram-creation.php
• Overlap: Sliver polygons
• Gaps
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25. Point feature topological errors
• Each polygon requires a label point to link it to
its attribute data.
• If a polygon has zero or several label points, an
error occurs.
×
11/7/2013 12:34 PM
×
25

26. Error sources
• Due to digitization errors within a data layer
– E.g., ‘Heads-down’ manual
map digitization using
a digitizing table.
– ‘Heads-up’ manual digitization
of CAD or scanned rasters
on the screen.
11/7/2013 12:17 PM
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27. Error sources
• Due to digitization or alignment errors
between layers.
• Don’t need to be of the
same feature type. E.g.:
– Adjoining layer edges don’t match.
11/7/2013 12:15 PM
– Distortion between
overlapping layers.
– Overlap, undershoot,
27
overshoot.

28. Building topology in ArcGIS
• Create topology
1. Define participating feature classes
2. Specify the topology rules between feature
classes
3. Set cluster tolerance
4. Identify the accuracy ranks of the coordinates in
each feature class.
5. Validate topology
6. Find and fix errors or mark exceptions
7. Iterate
11/7/2013 1:06 PM
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29. ArcGIS topology rules
• Topology rules allow you to define spatial
relationships between features in a single
feature class or subtype or between two
feature classes or subtypes.
• Topology rules allow you to define the spatial
relationships that meet the needs of your data
model.
• A pdf poster with all rules:
http://help.arcgis.com/en/arcgisdesktop/10.0
/help/001t/pdf/topology_rules_poster.pdf
11/7/2013 1:18 PM
29

30. Examples: ArcGIS topology rules
• Data theme: Parcels
• Feature classes:
– Parcel polygons,
– Parcel boundaries (lines)
– Parcel corners (points)
• Some topology rules
– Parcel polygons must not overlap.
– Parcel polygon boundaries must be covered by Parcel
boundary lines.
– Parcel boundary endpoints must be covered by Parcel
corner points.
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31. Identifies errors in Arc-Node topology
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32. Identifies polygon containment errors
Single
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33. Identifies adjacency errors
Single
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34. Identifies point in polygon
containment errors
Multiple
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34

35. Cluster tolerance
• ArcGIS analyzes the coordinate locations of
feature vertices among features in the same
feature class as well as between the feature
classes that participate in the topology.
• Those that fall within a specified distance of
one another are assumed to represent the
same location and are assigned a common
coordinate value (in other words, they are
collocated).
11/7/2013 1:22 PM
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36. Cluster tolerance
• A cluster tolerance is used to integrate
vertices.
– All vertices that are within the cluster tolerance
may move slightly in the validation process.
• The default cluster tolerance is based on the
precision defined for the dataset.
• The default cluster tolerance is 0.001 meters
in real-world units.
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37. XY tolerance
11/7/2013 1:28 PM
http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#//006200000003000000.htm
37

38. Z-tolerance
• If necessary, a z-tolerance is also defined to
distinguish whether or not the z-heights or
elevations of vertices are within the tolerance
of one another and should be clustered.
11/7/2013 1:28 PM
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39. Coordinate accuracy ranks
• Which vertices should move?
• Coordinate ranks ensure that reliably placed
vertices are the anchor locations toward which
less reliable vertices are moved. E.g.,
– RTK-GPS vertices would have high rank (1)
– Standard GPS vertices would have lower rank (2, 3)…
• Equally ranked vertices are geometrically
averaged.
11/7/2013 1:32 PM
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40. Validate topology: algorithmic
• Algorithmic
clustering and
cracking.
• Insertion of
common coordinate
vertices into
coincident features
that share geometry.
11/7/2013 1:59 PM
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41. Validate topology: manual
• Topology errors are violations of the rules that
are initially stored as errors in an error log.
• Error features record where topological errors
were discovered during validation.
– Example: Error features for the “must not have
dangles” rule.
11/7/2013 1:43 PM
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42. Fixing errors
• User intervention
– Change vertex locations, trace new lines and
replace old ones, etc.
Fix!
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43. Mark exceptions
• Certain errors may be acceptable, in which
case the error features can be marked as
exceptions. E.g.:
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44. Iterate
• Topology validation is user driven.
• Dirty areas are areas that have been edited,
updated, or affected by the addition or
deletion of features; they are maintained
automatically by ArcGIS.
• This allows selected parts, rather than the
whole extent of the topology, to be validated.
11/7/2013 1:50 PM
44